In this paper, we investigate the factor properties and gap sequence of the Tri-bonacci sequence, the fixed point of the substitution σ(a, b, c)=(ab, ac, a). Let ωp be the p-th occurrence of ω and Gp(ω) be the gap between ωp and ωp+1. We introduce a notion of kernel for each factor ω, and then give the decomposition of the factor ω with respect to its kernel. Using the kernel and the decomposition, we prove the main result of this paper:for each factor ω, the gap sequence {Gp(ω)}p≥1 is the Tribonacci sequence over the alphabet {G1(ω), G2(ω), G4(ω)}, and the expressions of gaps are determined completely. As an appli-cation, for each factor ω and p∈N, we determine the position of ωp. Finally we introduce a notion of spectrum for studying some typical combinatorial properties, such as power, overlap and separate of factors.
